Evidence of several dipolar quasi-invariants in Liquid Crystals
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Evidence of several dipolar quasi-invariants in Liquid Crystals
C.J. Bonin a,b , C.E. Gonz´alez a,c , H.H. Segnorile c,a and R. C. Zamar a,c a Facultad de Matem´atica, Astronom´ıa y F´ısica, Universidad Nacional de C´ordoba (FaMAF)M.Allende y H. de la Torre - Ciudad Universitaria, X5016LAE - C´ordoba, Argentina b Instituto de desarrollo tecnolgico para la industria - CONICET - Santa Fe, Argentina c Instituto de Fsica Enrique Gaviola - CONICET - C´ordoba, Argentina. (Dated: June 30, 2018)In a closed quantum system of N coupled spins with magnetic quantum number I , there areabout (2 I + 1) N constants of motion. However, the possibility of observing such quasi-invariant(QI) states in solid-like spin systems in Nuclear Magnetic Resonance (NMR) is not a strictly exactprediction. The aim of this work is to provide experimental evidence of several QI, in the protonNMR of small spin clusters, besides those already known Zeeman, and dipolar orders (strong andweak). We explore the spin states prepared with the Jeener-Broekaert pulse sequence by analyzingthe time-domain signals yielded by this sequence as a function of the preparation times, in a varietyof dipolar networks. We observe that the signals can be explained with two dipolar QIs only withina range of short preparation times. At longer times the time-domain signals have an echo-likebehaviour. We study their multiple quantum coherence content on a basis orthogonal to the z-basisand see that such states involve a significant number of correlated spins. Then we show that thewhole preparation time-scale can only be reconstructed by assuming the occurrence of multiple QIwhich we isolate experimentally. I. INTRODUCTION
Quasi-equilibrium states (QE) found in Nuclear Magnetic Resonance (NMR) of strongly interacting nuclear spinsystems are quantum states which do not evolve under the system Hamiltonian, thus they can be represented by areduced spin density operator diagonal in blocks in the eigenbasis of the spin-environment Hamiltonian [1, 2]. Thesestates only evolve due to spin-lattice relaxation, over a time scale much longer than the one of the build-up of the quasi-equilibrium. An experimental NMR procedure to prepare and detect dipolar
QE states in high-field solid-state NMRis the Jeener Broekaert pulse sequence (JB) [3]. Briefly, this technique consists of the phase-shifted radiofrequency(rf) pulses: 90 x − τ − y − t e − y − t . The first pulse creates single quantum coherences in the spin system, whichevolve in the rotating frame mainly under the dipole spin-spin Hamiltonian during τ . Along this period, multi-spinsingle-quantum coherences can develop, and the second 45 y pulse transforms part of the coherences just created intomulti-spin order [2, 4, 5]. Two important processes occur during the evolution period t e , along different timescales:decoherence and relaxation. Finally, the third pulse converts QE states into observable single quantum coherence.JB experiments are frequently used to transfer the Zeeman order into dipolar order, which, in a variety of samplesis the only observable dipolar QE state. It is an experimental fact that for short preparation times τ ≪ /ω D inthe JB sequence, the dipolar signal observed after the third pulse is proportional to the time derivative of the FIDsignal [12], which indicates that the prepared state is very similar to the secular (high field) dipolar energy. Twodipolar QE states, were instead observed both on hydrated salts [6, 7], and on liquid crystals (LC) [8, 9]. In theformer, the occurrence of ‘intra-pair’ and ‘inter-pair’ QE states was associated with the distribution of the protonpairs of the water molecules in the lattice, conforming a system of weakly interacting spin pairs. The spin systemof LCs is composed by the few proton spins within each molecule, which display a hierarchy of dipolar couplings.This kind of dipolar network is different from hydrated salts and a model of spin pairs does not strictly proceed,however, the dipolar Hamiltonian can in fact be partitioned into two mutually commuting and orthogonal parts, the‘strong’ H S and the ‘weak’ H W terms [10]. Since these operators are chosen so that they commute with the totalspin Hamiltonian, they are nearly constants of the motion (for evolution times t e much greater than 1 /ω D [11]). Forthis reason, they are called quasi-invariants (QI).In early works on dipolar order relaxation in LC’s, the experiment was interpreted in terms of a single quasi-invariant, ρ S , of an ensemble of isolated representative strongly coupled spin pairs [13]; however, at that time,the dependence of relaxation times on τ was not understood [14]. Later work on LC showed that the state ρ W prepared by setting the preparation time to the one at which the ρ S order vanishes, also presents properties of quasi-equilibrium. In fact, it relaxes exponentially towards equilibrium, with a characteristic time which is also of the order T D [9]. Besides, experiments where the multiple quantum coherences of the states prepared with the JB sequenceare encoded in orthogonal, X basis, carried out in nematic 5CB [15], confirmed that the tensor structure of the S state corresponds to two-spin dipolar order while the W state shows high order quantum coherences, all of whichrelax towards equilibrium with the same decay constant T W . This indicates that for short preparation times thetwo-spin order prevails while higher order spin correlations are growing. This second QI, produces discernible signalscoresponding to multi-spin order, created within a time window τ where the dipolar order vanishes, similarly to thehydrated crystal studied in reference [7].Recently, it was shown that quantum decoherence can provide an efficient mechanism by which the spin systemattains a diagonal state in the basis of the spin-environment Hamiltonian, over a time scale which is intermediatebetween those governed by its own interactions, and thermalization ruled by thermal fluctuations of the environment[1, 16]. All these results support the assumption that in LC’s, for times t e greater than the decoherence time scale,the density operator can be written in the form [10] ρ D = − β Z H Z − β S ( τ ) H S − β W ( τ ) H W . (1)In fact, the experimental behaviour of the dipolar signal of 5CB can be described with Eq.(1) for preparation times τ ≤ µ s [9]. Relaxation brings the system to thermal equilibrium with the whole surrounding world over a much longertime scale, t e ≫ τ , through processes involving energy exchange between the spins and the lattice. The characteristiclifetime T D of QE states (dipolar order relaxation time) is comparable to the common Zeeman relaxation time.Theoretically, in a cluster of N dipole interacting spins 1/2, as the protons of a typical LC molecule, there are atleast 2 N constants of motion, the exact number depends on the degeneration of the dipolar Hamiltonian [2]. Thatis, such a number of spin operators are needed to span the commutative space (or diagonal in blocks space) of thedipolar Hamiltonian. The results obtained so far: the occurrence of two dipolar QI, are consistent with this idea, butaccordingly, it should be possible to prepare new quasi-equilibrium states from the initial Zeeman order, which shouldbe observable for longer preparation times in the JB experiment, as long as the JB signal is detectable.The aim of this work is to explore the spin states prepared for long preparation times in the JB experiment, lookingfor experimental evidence of multiple QIs that can be expected for dipole coupled spin clusters. The analysis is basedon an exhaustive study of the time-domain signals in a variety of dipolar networks.Providing insight on the physics nature of the quasi-invariants of a spin cluster can be useful both for applicationsas for basic research. QE states are relevant observables of the spin system, providing relaxation parameters usefulto study molecular motion in LC mesophases through their dependence on temperature and magnetic field [9, 17, 18]They have been proposed as initial states for the excitation of multiple quantum coherences (MQ) in MQ-NMR [19–21]and used in spin counting experiments [5, 15]. Besides, these states have been proposed as alternative to the Zeemanorder in Magnetic Resonance Imaging [22]. Implementation of noiseless quantum memories and multi-spin quantumregister relies on the possibility of manipulating multi-spin correlated entities which are unperturbed by decoherenceprocesses [23–25, 28]. From a basic viewpoint, the physics of systems with few degrees of freedom coupled to a quantumenvironment attracts today’s attention of a widespread community because of the potentiality for applications suchas quantum devices and quantum information processing [26–28] and, significantly, also because these systems aretestbeds for studying fundamental aspects as irreversibility, equilibration and thermalization [1, 29, 30].Section II contains the definition of the QI operators and their relation with the NMR signals in the JB experiment.In Section III we present an experimental survey of the manifestation of multiple quasiinvariants and propose amethod to isolate them in samples with different geometries. The nature of the different QI is examined by means ofrelaxation experiments and by studyng the multiple quantum coherence content on the x-basis. II. QUASI INVARIANTS
The spin Hamiltonian of resonant nuclei in ordered systems like solids and liquid crystals has an important contri-bution from the dipole-dipole coupling energy besides the Zeeman term, H S = H Z + H D . (2)where the Zeeman energy in units of ¯ h is H Z = − ω o I z , with ω o the Larmor frequency. In the rotating frame description[31], the time evolution of any spin state is mainly driven by the dipolar Hamiltonian and often only by its secularpart (high field approximation), H D = √ X i The interest herein is to explore the manifestation of multiple QIs after the Jeener-Broekaert rf pulse sequence.With this aim we analyze the JB signals as a function of the preparation time in different kinds of dipolar networks,with particular attention on long preparation times. Experiments at 20MHz were carried on in a Bruker Minispecmq20 and those at 60MHz in a homemade pulsed NMR spectrometer based on a Varian EM360 magnet [34, 35].As shown in Fig.1, the response of the different compounds to variations of the preparation time is diverse, howeverthey all share the characteristic of being symmetric with respect to τ and t . In powder adamantane (Fig.1a), thesignal is proportional to the time derivative of the FID and keeps this shape for every preparation time τ , which isconsistent with the occurrence of only one dipolar constant of motion: the dipolar energy, (cid:10) H D (cid:11) [3, 36].On the contrary, in the other compunds shown in Fig.1 [(b) Gypsum and (c) N-(4-Methoxybenzylidene)-4-butylaniline (MBBA)], the dipolar signal shapes change drastically with τ , being proportional to the time derivativeof the FID only for short τ (more precisely, the proportionality holds while τ < t , where t is the time at which thetime derivative of the FID first crosses through zero).However, a new behaviour of the dipolar signals arises clearly in LC for longer preparation times. The observedsignals show an “echo” like behaviour, that is, they have a peaked shape whose maximum occurs at a time t from theread pulse equal to the preparation time τ . Such behaviour is not observed in gypsum.In order to highlight this feature, Figs. 2 and 3 show the time t max at which the dipolar signal attains its maximumamplitude in the detection period, as a function of the preparation time τ , in different compounds. Preparationtimes were varied within the range where the S/N ratio is greater than 1% on each compound. It can be seen thatthe experimental t max ( τ ) (solid circles in Fig. 2) is very similar in all the cyanobiphenyl samples, 4’- n -4-biphenyl-carbonitrile where n stands for pentyl (5CB), hexyl (6CB) and octyl (8CB). Data from nematic MBBA at the bottomof Fig. 3 behave just as the other nematic LC samples. This behaviour is drastically different in the gypsum singlecrystal (top of Fig. 3) since t max ( τ ) is bounded in gypsum while grows linearly for long τ in all the nematic LCsamples. As seen in this figure, the whole observable range on gypsum is rather smaller than that of the LCs, and theremarkable difference is that LCs develop the slope-one behaviour at preparation times near one fourth of their wholeobservable ranges while gypsum never does. It is also worth to note that the behaviour of t max ( τ ) is also differentin two orientations of the gypum single crystal with respect to the external magnetic field. The wavy curve of solidcircles in Fig. 3(top) corresponds to the orentation ~B k [010] while the open squares correspond to an orientationthat makes the protons at the water molecules equivalent.It has been already shown that two independent QI ( S and W ) can be prepared in nematic LC and some hydratedsalts [9, 37], with the property that W emerges when the weight of S becomes zero. In fact, the set of dipolarsignals in 5CB prepared with τ < µ s were satisfactorily reproduced by using such two QI in Eq.(5) [9]. However,this procedure does not describe the signal behaviour for τ > µ s. Now, we propose a procedure to extract thequasiinvariants from the experimental data, under the sole assumption that the states prepared in the experimentare described by Eq.(4). It might be expected that new QI, if observable, will arise sequentially as the preparationtime increases and that in a favourable condition, a subset of preparation times { τ i } exists at which one QI (the i -th)prevails over the others. This amounts to proposing that it is experimentally possible to isolate a subset of QI. Thevalidity of such working hypothesis will be tested along this work.The measured data sets are composed by the signals obtained for each of the m chosen values of the preparationtime τ = τ m . Let us call M τ m ( t ) to the acquired time domain signals after the JB sequence, which we arrange asthe rows of the data array, and call N t ( τ ) to the columns of such array (pseudo-signals). Any cut N t ( τ ) at t = t isfound to be strictly symmetric with the signal M τ ( t ) prepared with τ = t .With the aim of deriving the functions F i ( t ) from the experimental data we followed these steps: ( i ) Recognize thepreparation time τ = τ which makes M τ ( t ) proportional to the time derivative of the FID signal and has maximumamplitude. This signal attains its maximum at t = t = τ . Based on former evidence that the state prepared with atime τ = τ is a QI, we assume that the experimental function M τ ( t ) can be identified with the contribution of thefirst QI to the signal, F ( t ) (see dotted line in Fig. 4). ( ii ) Also select the experimental pseudo-signal N t ( τ ), whichcorresponds to a cut on the data array at an observation time t = t , and identify it with the “weight” β ( τ ) of thefirst QI. If the only dipolar QI is the dipolar energy, as is the case of adamantane, the dipolar signal of Fig. 1(a) canbe adequately reconstructed by calculating the 2D function S ( τ, t ) = β ( τ ) F ( t ) ≡ N t ( τ ) M τ ( t )However, this form of S ( τ, t ) does not describe the experimental signals of the other studied samples, then, ( iii ) Inorder to select the contribution of the next QI, F ( t ), we analyze the difference between the measured data and thecontribution of the first QI to the signal C ( τ, t ) = M τ ( t ) − S ( τ, t ) . This data set attains its maximum value at a time which we call t = τ . Then we choose F ( t ) ≡ C ( τ , t ) and β ( τ ) ≡ C ( τ, t ), the solid curve in Fig. 4. It is worth to mention that N t ( τ ) ≃ 0, and therefore τ is the time atwhich the the first dipolar QI has negligible contribution to the observed signal (This feature agrees with previousworks [9, 15, 37]).This election of F and F for reconstructing the NMR signal S ( τ, t ) yields the curves in open triangles in Figs.2 and 3 (blue triangles on the online version). It allows for an excellent reconstruction of the experimental curvein gypsum. In fact, as seen in the top of Fig.3 both curves (solid circles and open triangles) coincide in almost allthe timescale. The dipolar signal shape had already been accounted for within a restricted interval of preparationtimes by Dumont et al. [37] by assuming the occurrence of two QI. Now we learn that these two QI can give a gooddescription within an extended timescale.On the contrary, in LC samples, the triangle-curves only reproduce the experimental behaviour of t max ( τ ) for times τ, t < µ s in 5CB, 6CB 8CB and MBBA, that is, when the preparation times are restricted roughly to within the firsthalf period of the dipolar coupling. This failure in describing t max ( τ ) for longer times with only two quasiinvariantsled us to continue with the procedure in order to find a new curve F . Again we analyze the residue C ( τ, t ) = M τ ( t ) − S ( τ, t )to find the time t = τ at which this data set attains its maximum amplitude and define F ( t ) ≡ C ( τ , t ) (seedashed curve in Fig. 4) and β ( τ ) ≡ C ( τ, t ). The times t , t and t corresponding to the different compounds aresummarized in the following Table. Compound Temp (K) t ( µ s) t ( µ s) t ( µ s)5CB 302 30 70 1345CB 297 26 60 1066CB 297 36 85 1608CB 311 29 74 140gypsum 311 10 28 -MBBA 311 22 65 120With this selection it was possible to improve the reconstructed curve for t max ( τ ). Figures 2 (a,b,c) and 3 (b) showthat involving a third quasiinvariant yields the open circles curve which gives a noticeably better reconstruction ofthe experimental curves. Fig.5 shows a detail of t max ( τ ) in MBBA, where the three-QI curve is definitely better thanthe two-QI curve especially within the interval 115 µ s < τ < µ s; however it still does not account for the echo-likebehaviour for longer preparation times.In order to test if F ( t ) does in fact behave as the signal of a QI, we analyze the signal amplitude attenuationas a function of the evolution time t e as in a regular spin-lattice relaxation experiment. Figure 6 shows amplitudeattenuation of F ( t ) in 5CB at 302 K, for the preparation times shown in Table 1: τ = 30 µ s, circles; τ = 70 µ s, opensquares and τ = 134 µ s, full squares. The three curves can be adequately fitted with two exponential decays, thecommon short time decay (the same for all) is 4 ms while the longer characteristic times are T D = 67 ms (circles), T D = 56 ms (open squares) and T D = 39 ms (full squares). The fast decay is just a witness of the attenuationof higher order coherences, in fact, its characteristic time coincides with the timescale of irreversible decoherence asmeasured in ref. [16]. On the other hand, the longer decay times are all of the order of the dipolar relaxation time (67 ms in this case).In ref. [15] it was shown that the quasi equilibrium state which is proportional to H S in 5CB (prepared by setting β W ( τ ) ≃ 0) is a two-spin correlated state since it involves at most two-spin tensors, while states proportional to H W (prepared by setting β S ( τ ) ≃ 0) involve correlations of more-than-two spin tensors. With the aim of studying thecorrelated nature of the states prepared at different τ ’s in 5CB, we analyze the multiple quantum coherence (MQc)content on the X-basis using the same pulse sequence as in refs. [5, 15]. In this experiment, rotating the state aroundan axis orthogonal to Z allows encoding MQc which reflect the number of multiply connected spins in the preparedstates. The experiment begins by preparing the state with the phase shifted JB pulse pair, then, a waiting time t w isfollowed by two pulses 90 ( φ + π/ - ǫ - 90 ( y ) which encode the coherence numbers of the quantum state at time t w inthe X-basis, when varying φ systematically in succesive experiments. Fourier transformation of the signal amplitudewith respect to φ yields an X-basis coherence spectrum. Our experiment, conducted on 5CB at 297 K and 60 MHz,was set to encode up to 8-quantum coherences on the X-basis, however, coherences higher than 4 in 5CB were seen tofall below the noise level. The MQc content varies with the preparation time τ , as shown in Fig. 7 (a). As expected,zero and double quantum coherences are dominant for 0 < τ < µ s (notice that t i in 5CB change with temperature,as evident in the first two rows of Table 1). The fact that the quotient of their amplitudes is ca. 1.5 indicates thatthe states prepared within this interval can be described with only one QI (or dipolar constant of motion) [5], whichis proportional to T , the 0 component of a spherical tensor of rank 2. Coherences ± τ and dominate only within restricted intervals.At a preparation time near 60 µ s the ratio between coherences 2 and 0 changes drastically and their amplitudesbecome comparable to the rising 4-quantum coherence. Notice that this is precisely the preparation time at which C ( τ , t ), the signal associated with the second QI, attains its maximum, showing that these states involve correlationbetween more-than-two spins. This characteristic indicates that these are multi-spin-correlated states. Notice thatalso near 110 µ s all the coherences attain comparable amplitudes, and this preparation time coincides with the thirdQI, F .We observed that the relative amplitudes of the various MQc components depend on which window along theacqusition time is used for calculating the MQc spectrum. This feature is shown in Fig.7(b) where the time interval(along the acquisition period) at which the maximum value of each coherence component occurs is plotted as afunction of the preparation time. The salient feature is that different coherences have a very different behaviour.The maximum contribution to coherences 0 and ± t max = 30 µ s, exceptwithin the narrow preparation intervals which correspond to the second and third quasiinvariants. On the contrary,the maximum amplitude of coherences ± t max = τ , in other words, it behaves likean echo. This distinct behaviour agrees with a view that multi-spin correlations arise at longer times than two-spincorrelations, and also that the echo like signals prepared with τ > µ s correspond to states of multi-spin nature. IV. DISCUSSION AND CONCLUSIONS In this work we present an experimental survey on the occurrence of several QI created using the JB sequence, indifferent dipolar-coupled spin systems. To account for the dependence of the JB signals on the preparation time, wepropose a generalization of the method presented in refs.[9, 37]. We demonstrate that multiple QI can be preparedin the dipole interacting proton spin system on LC molecules. The well-known dipolar order state, which is a two-spin object, comes at short preparation times, while the other QI which emerge sequentially at longer preparationtimes involve multiple-spin interactions. We propose a criterion to experimentally isolate each QI and oberve thatthe first three states in 5CB have evidently distinct spin-lattice relaxation times which implies that they are truequasiinvariants.Though at present the tensor structure of these quasi-invariants is still unknown, the multiple quantum coherenceexperiments demonstrate that the new states have multi-spin character, which implies that their tensor structuremust involve products of individual angular momentum operators of many spins. Also, we verified that the observedecho-like behavior of the dipolar signal in LC for long preparation times is also a consequence of the occurrenece ofmany QI.In this way we demonstrate that the initial Zeeman order (of thermal equilibrium) can in fact be transferred to atleast three dipolar constants of motion in LC (within the scanned short and intermediate τ range), while only twoconstants of motion were observed in gypsum when ~B k [010] and only one at an orientation that makes the protonsat the water molecules equivalent. The fact that compounds having very different numbers of interacting spins (11 to25 in LC and infinite in gypsum) admit the preparation of more than one dipolar QI, indicates that their occurrenceis not exclusively determined by the size of the dipolar network. On the contrary, both the occurrence of multiple QIand the many-spin character they show, do depend strongly on the topology of the spin network, since it determinesthe quantum dynamics of the spin system. This assertion also agrees with the fact that the quality of our three QImodel description varies from one LC to another.It is worth to note that the feasibility of transferring the Zeeman order to multi-spin quasi-equilibrium statesis not an exclusive feature of spin clusters as the proton system in LC’s as illustrated in this work. In fact, the“interpair” state observed in hydrated crystals admits multiple quantum coherences up to the fourth order on the X − basis, implying four-spin correlations [15]. Then the question arises about which are the necessary characteristicsthe dipolar network must have in order to admit the preparation of more-than-one dipolar QI. A coarse criterioncould be the feasibility of truncating the weaker term of the dipolar Hamiltonian with respect to the stronger term,which generally implies the possibility of clasifying the dipolar couplings according to their intensity into “strong”and “weak”. In practice, this characteristic is also associated to the occurrence of a doublet in the NMR spectrum.According to our results, in LC’s two-spin correlations dominate the coherent dynamics during the early timescaleof the preparation period, however, there are narrow time windows where these correlations vanish and higher or-der correlations can efficiently give place to multi-spin QE states. Finally, for longer τ values only the multi-spincorrelations are responsible for the echo-like behavior of the NMR signal. [1] H.H. Segnorile and R.C. Zamar, J. Chem. Phys. 135, 244509 (2011).[2] J.D. Walls and Y. Lin, Solid State Nuclear Magnetic Resonance, , 22 (2006).[3] J. Jeener and P.Broekaert, Phys. Rev., , 232 (1967).[4] J. Baum, M. Munowitz, A.N. Garroway and A. Pines, J. Chem. Phys., 83, 2015 (1985).[5] H. Cho, D. Cory and C. Ramanathan, J. Chem. Phys., 118 (8), 3686 (2003).[6] H. Eisendrath, W. Stone, and J. Jeener, Phys. Rev. B, , 47 (1978).[7] A. Keller, Adv. Magn. Reson., 12 183 (1988).[8] H. Schmiedel, S. Grande and B. Hillner, Phys. Letters A, 365 (1982).[9] O. Mensio, C.E. Gonz´alez and R.C. Zamar, Phys. Rev. E, 71, 011704 (2005).[10] H.H.Segnorile, C.J.Bonin, C.E.Gonz´alez, R.H.Acosta, R.C.Zamar, Solid State Nucl. Magn. Reson. (2009).[11] D.P. Weitekamp(2008). Time Domain Multiple Quantum NMR. Lawrence Berkeley National Laboratory. Retrieved from:http://escholarship.org/uc/item/40c6h0zq[12] The NMR signal ensuing a single, radio frequency pulse is called FID for Free Induction Decay signal.[13] R.G.C. Mc. Elroy, R.T. Thompson and M.M. Pintar, Phys. Rev. A 10, 403 (1974).[14] F. Noack, St. Becker and J. Struppe, Annual Reports on NMR Spectroscopy, 33, 136 (1996).[15] L. Buljubasich, G. A. Monti, R. H. Acosta, C. J. Bonin, C. E. Gonzlez, and R. C. Zamar, J. Chem. Phys. 130, 024501(2009).[16] C. E. Gonzlez, H. H. Segnorile and R. C. Zamar, Phys. Rev. E 83, 011705 (2011).[17] R. Zamar, E. Anoardo, O. Mensio, D. Pusiol, S.Becker and F. Noack, J. Chem. Phys. 109, 3, 1120 (1998).[18] O, Mensio, R. Zamar, E. Anoardo, R. H. Acosta, and R. Y. Dong, J. Chem. Phys. 123, 204911 (2005).[19] S. Emid, J. Smidt, and A. Pines, Chem. Phys. Lett. 73, 496 (1980).[20] S. I. Doronin, E. Feldman, E. Kuznetsova, G. B. Furman, and S. D. Goren, Phys. Rev. B 76, 144405 (2007); Prisvma vZhETF 86, 26 (2007).[21] B. Furman and S. D. Goren, J. Phys.: Condens. Matter 17, 4501 (2005).[22] S. Matsui, S. Saito, T. Hashimoto, and T. Inouye, J. Magn.Reson. 160, 13 (2003).[23] H. G. Krojanski and D. Suter, Phys. Rev. Lett. 93, 090501 (2004); Phys. Rev. Lett. 97, 150503 (2006).[24] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998).[25] A. K. Khitrin, V. L. Ermakov, and B. M. Fung, Chem. Phys. Lett. 360, 161 (2002).[26] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cam-bridge,2001).[27] D. Suter and T. S. Mahesh, J. Chem. Phys. 128, 052206 (2008)[28] T. S. Mahesh and D. Suter, Phys. Rev. A 74, 062312 (2006).[29] V.I. Yukalov, Physics Letters A 376, 550 (2012).[30] A. Polkovnikov, K. Sengupta, A. Silva, M. Vengalatore, Rev. Mod. Phys. 83, 863 (2011).[31] Malcolm H. Levitt Spin Dynamics, Basics of Nuclear Magnetic Resonance (J. Wiley & Sons, Chichester 2005).[32] A.Abragam, The Principles of Nuclear Magnetism (Oxford U.P. London 1961).[33] A. Abragam and M. Goldman, Nuclear Magnetism: order and disorder , 054427 (2005).[37] E. Dumont, J. Jeener and P.Broekaert, hys. Rev. B, , 6764 (1994).[38] J. Jensen, Phys. Rev. B, 52, 9611 (1995). t ( s) t ( s) t ( s) FIG. 1: Stackplots of the JB signals as a function of the preparation time τ in (a) Adamantane, (b) Gypsum and (c) MBBAliquid crystal. t m a x ( s ) ( s) t m a x ( s ) ( s) t m a x ( s ) ( s) FIG. 2: Acquisition time t max at which the signals attain their maximum amplitude as a function of the preparation time τ in the nematic phase of 5CB (top), 6CB (middle) and 8CB (bottom). Solid circles represent the experimental data. Opentriangles and open circles correspond to t max ( τ ) simulated from Eq.(5) with two and three quasiinvariants respectively. QIsselected as indicated in Table 1. t m a x ( s ) ( s) t m a x ( s ) ( s) FIG. 3: Acquisition time t max at which the signals attain their maximum amplitude as a function of the preparation time τ ina gypsum single crystal (top) and in nematic MBBA (bottom). Solid circles represent the experimental data. Open trianglesand open circles correspond to t max ( τ ) simulated from Eq.(5) with two and three quasiinvariants respectively. QIs selectedas indicated in Table 1. Solid circles on the gypsum graph correspond an orientation ~B k [010], while open squares to anorientation that makes the protons at the water molecules equivalent. a m p li t ude ( a r b . ) t ( s) FIG. 4: Signals F (dotted), F (solid) and F (dashed) for 5CB at 302K. 100 200100200 t m a x ( s ) ( s) FIG. 5: Detail of the experimental t max ( τ ) in nematic MBBA (solid circles) and its description with Eq.(5) considering two QI(open triangles) and three QI (open circles). (cid:13) a m p li t ude ( a r b . ) (cid:13) t(cid:13) (ms)(cid:13) FIG. 6: Spin-lattice relaxation curves of three QI in 5CB at 302K prepared with different preparation times: τ = 30 µ s, circles; τ = 70 µ s, open squares and τ = 134 µµ